Question

Step 3: write an equivalent expression for sin4x that does not contain powers of trigonometric functions greater than 1.

Answer 1

Answer: I don't know if you wanted to write sin(4x) or $sin^{4} (x)$ , but here we go:

ok, sin(4x) = sin(2x + 2x), and we know that:

sin (a + b) = sin(a)*cos(b) + sin(b)*cos(a)

then sin (2x + 2x) = sin(2x)*cos(2x) + cos(2x)*sin(2x) = 2cos(2x)*sin(2x)

So 2*cos(2x)*sin(2x) is equivalent of sin(4x)

If you writed $sin^{4} (x)$ then:

$sin^{4} (x) = sin^{2} (x)*sin^{2} (x)$

and using that: $sins^{2} (x) = \frac{1-cos(2x)}{2}$

we have: $sin^{2} (x)*sin^{2} (x) = \frac{(1-cos(2x))*(1-cos(2x))}{4} = \frac{1-2cos(2x) + cos^{2}(2x) }{4}$

and using that: $cos^{2} (x) = \frac{1 + cos(2x)}{2}$

$\frac{1-2cos(2x) + cos^{2}(2x) }{4} = \frac{1-2cos(2x) + \frac{1+cos(4x)}{2} }{4}$

You can keep simplifying it, but there is your representation of $sin^{4} (x)$  that does not contain powers of trigonometric functions greater than 1.